Soil moisture dynamics is a key factor controlling hydrologic fluxes through the soil–plant–atmosphere continuum (SPAC). Thus, soil moisture dynamics can be fundamental information for describing ecosystem processes such as water stress and carbon assimilation (e.g. Rodriguez-Iturbe *et al*. 1999; Laio *et al*. 2001; Porporato *et al*. 2001; Porporato, Daly & Rodriguez-Iturbe 2004; Rodriguez-Iturbe & Porporato 2004). Accounting only for changes in mean responses to climatic variability is not sufficient for realistic investigations on the effects of climate change on ecosystems. Instead, such investigations must account for the stochastic component of hydrologic forcing and its possible changes in terms of frequency and amount of rainfall events. These changes are responsible for modifying soil moisture dynamics and the temporal structure (i.e. intensity, duration and frequency) of periods of water stress and impaired plant assimilation (Porporato *et al*. 2001, 2004). Probabilistic descriptions of ecosystem process dynamics are useful to describe the effects of climatic fluctuations such as severe drought induced by current ENSO cycles and global climate change.

In this study, we used a simple SPAC model linked to probabilistic descriptions of soil moisture and plant water stress (Porporato *et al*. 2001), to arrive at a synthetic characterization of plant carbon assimilation dynamics (Daly, Porporato & Rodriguez-Iturbe 2004). The computed results were linked to previous observations in a tropical rainforest in Southeast Asia (Kumagai *et al*. 2004a,b,c, 2005, 2006; Manfroi *et al*. 2006; Ohashi *et al*. 2008). We compared the dynamics of plant water stress and productivity between isohydric and anisohydric species in various rainfall regimes. We used a stochastic representation to test their advantages and survival in the studied tropical rainforest site under the current climatic conditions and under possible climate change-induced drought. For completeness, a review and description of the modelling scheme is presented in the following discussion (see Rodriguez-Iturbe & Porporato 2004) and a list of symbols used in the model is given in the Appendix.

#### Stochastic soil moisture dynamics model

The stochastic soil moisture model used in this study was originally proposed by Rodriguez-Iturbe *et al*. (1999) and improved by Laio *et al*. (2001). Under the conditions that lateral movement can be neglected, the vertically integrated soil moisture balance equation is given as follows:

- (1)

where *n* is the soil porosity, *Z*_{r} is the rooting depth, *s* is the degree of saturation varying between 0 and 1, *t* is time, *R* is the rainfall rate, *I* is the canopy interception loss, *Q* is the rate of infiltration-excess, *E*(*s*, *t*) is the evapotranspiration rate and *L*(*s*) is the leakage. Equation 1 is a stochastic ordinary differential equation for the state variable *s*, and is solved on daily timescales. The frequency of rainfall events can be assumed to be a stochastic variable expressed as an exponential distribution with the mean time interval between precipitation events 1/*λ*. The amount of rainfall, when it occurs, is also assumed to be an independent random variable described by an exponential probability distribution with the mean depth of rainfall events, *α*. Losses to the atmosphere from canopy interception, *I*, are represented using a threshold of rainfall depth Δ, which denotes interception capacity and below which effectively no water reaches the ground. Saturation overland flow, *Q*, is derived from rainfall excess, which occurs when rainfall depth exceeds the available storage.

Leakage losses are assumed to occur through gravitational flow. *L*(*s*) is assumed to be at its maximum for saturated soil moisture conditions and can be expressed as the hydraulic conductivity *K*(*s*), that is, the exponential function of *K*_{sat}, the saturated hydraulic conductivity, *s*_{fc}, *s* at ‘field capacity’, the water content held in the soil after gravity drainage where *K*(*s*) has a value of zero, and *β*, a parameter that can be obtained by fitting the power law unsaturated hydraulic conductivity to the field data (e.g. Laio *et al*. 2001).

In this model, the main limiting factor controlling the evapotranspiration is considered to be soil moisture. Specifically, the dependence on *s* is given by (e.g. Laio *et al*. 2001)

- (2)

where *E*_{W} and *E*_{max} are the soil evaporation and the maximum evapotranspiration, respectively, and *s*_{h}, *s*_{W} and *s** are *s* at ‘hygroscopic point’, ‘plant wilting point’ and ‘plant stress point’, respectively. When *s* falls below a given *s**, plant transpiration is reduced by stomatal closure to prevent internal water losses. Then, soil water availability becomes a key factor in determining the actual evapotranspiration rate. Transpiration and root uptake continue at a reduced rate until *s* reaches *s*_{W}. Below *s*_{W}, soil water is further depleted only by evaporation at a low rate to *s*_{h}.

When the soil contribution to total evaporation is insignificant (given the high leaf area index, LAI, at the site) and in the absence of free water on vegetation, the bulk surface conductance is strongly related to the behaviour of leaf stomatal conductance (Kelliher *et al*. 1995; Raupach 1995, 1998). Equilibrium evaporation occurs when air passes over an extensive wet surface and becomes saturated. However, because even such air is seldom saturated, the ratio of actual evaporation to the equilibrium evaporation is often 20–30% greater than unity (Priestley & Taylor 1972). This ratio is referred as the Priestley–Taylor coefficient, *α*_{PT}. De Bruin (1983) found that although *α*_{PT} varies with wind speed, surface roughness, and the entrainment of dry air at the top of the atmospheric boundary layer, it depends primarily on the surface conductance. Furthermore, McNaughton & Jarvis (1983) showed that equilibrium evaporation is the evaporation at the limit of complete decoupling. In our previous studies (Kumagai *et al*. 2004c, 2005), we found high decoupling coefficient values because of the relatively large surface conductance compared with aerodynamic conductance, and that net available energy is the primary forcing variable for evapotranspiration at the study site. Thus, we used a modified Priestley & Taylor (1972) expression to compute *E*_{W} and *E*_{max}, as follows:

- (3)

where *i* denotes *W* or max, *α*_{PT_W} and *α*_{PT_max} are the Priestley–Taylor coefficients (*α*_{PT}) for *E*_{W} and *E*_{max}, respectively, *κ* is a unit conversion factor, *δ* is the rate of change of saturation water vapour pressure with temperature, *L*_{v} is the latent heat of vapourization of water, *ρ*_{w} is the density of water, *ε* is the psychrometric constant, and *R*_{n} is the net radiation above the canopy. A major uncertainty in Eqn 3 is *α*_{PT}, which, for forested ecosystems, is usually less than its typical value of 1.26 because of additional boundary layer, leaf, xylem and root resistances. Measured evapotranspiration data are used to obtain daily *α*_{PT}, and then the relationship between *s* and daily *α*_{PT} can be derived. *α*_{PT_W} and *α*_{PT_max} are parameterized by fitting Eqns 2 and 3 to the relationship (see Fig. 2a). For the stochastic model computations, all thermodynamic variables in Eqn 3 were assumed to be constant values, *R*_{n} was also obtained by averaging yearly observed values, and the effect of atmospheric humidity on stomatal conductance was ignored to maintain analytical tractability. Note that using averaged *R*_{n} (which gave an averaged value of evapotranspiration; Kumagai *et al*. 2004b, 2009), and using a function for stomatal response to increasing vapour pressure deficit (Kumagai *et al*. 2004a,b) resulted in little change to the probabilistic descriptions of soil moisture and daily transpiration rate. Figure 2a summarizes the evapotranspiration model used in this study.

Because of the stochastic rainfall forcing in Eqn 1, its solution can be represented only in a probabilistic manner. In this framework, the probability density function (PDF) of *s*, *p*(*s*), can be derived from the master equation of the process (see Rodriguez-Iturbe & Porporato 2004).

#### Carbon assimilation and plant water stress dynamics

Daily carbon assimilation *A*_{n} is defined as the canopy photosynthetic rate, which does not take into consideration the carbon lost by ecosystem respiration. A behaviour similar to that of *E*(*s*, *t*) is also found for the dependence of *A*_{n} on *s* (see Fig. 2a,b). This dependence may be functionally approximated as follows (Daly *et al*. 2004):

- (4)

where *s*_{WA} and *s**_{A} are *s*_{W} and *s** for assimilation, respectively, and *A*_{W} and *A*_{max} are *A*_{n} at *s*_{WA} and the maximum value, respectively. The probabilistic structure of *A*_{n} can be derived from *p*(*s*) by a derived-distribution approach. Accordingly, the PDF of *A*_{n}, *p*(*A*_{n}), has an atom of probability in zero, ; one in *A*_{max}, ; and is continuous in between. In this study, the mean assimilation during a season, , is used to analyse productivity of isohydric and anisohydric plant species in each given climatic condition.

Following Porporato *et al*. (2001) the (static) plant water stress, *ζ*, can be defined as zero when soil moisture is above the level of incipient stomatal closure, *s**, and defined as the maximum value of one when soil moisture is at the level of complete stomatal closure, *s*_{W}. There is a non-linear curve in the value of *ζ* between *s** and *s*_{W}:

- (5)

where *q* is a measure of the non-linearity of the effect of soil moisture deficit on plant conditions, and can vary with plant species (Fig. 2c). Plants with a higher *q*-value, that is, stronger non-linearity, can have lower *ζ* with decreasing *s* but abruptly increase *ζ* in the immediate vicinity of *s*_{W}. This means that such plants can allow their leaf water potential to approach the xylem cavitation threshold without suffering from water stress (Porporato *et al*. 2001).

Like in the case of *p*(*A*_{n}), the probabilistic structure of *ζ* can be derived from *p*(*s*) by substituting Eqn 5 for *s*. The PDF of *ζ*, *f*_{Z}(*ζ*), has an atom of probability at *ζ* = 0, *F*_{Z}(0) = 1 − *P*(*s**), and one at *ζ* = 1 equal to *F*_{Z}(1) = *P*(*s*_{W}) (where *P* denotes the cumulative density function of *s*). The continuous part of *f*_{Z}(*ζ*) between zero and one corresponding to *s*_{W} < s < s* can be written as follows (Porporato *et al*. 2001):

- (6)

in which

- (7)

where *C*_{ζ} is an integration constant, which can be found by imposing the condition .

The mean water stress can be calculated as . However, as the value of also takes into consideration the periods of *ζ* = 0, we considerthe mean value of water stress given that, for our purposes, the plant is under stress (Porporato *et al*. 2001):

- (8)

where *P*(*s**) denotes only the part of the PDF corresponding to *ζ* above 0.

The information on plant conditions given by the mean and the PDF of water stress needs to be completed by describing the duration of the period of water stress. The length of the time in which soil moisture is below a threshold is among the most important stochastic variables to analyse the temporal dimension of water stress. The analytical expression for the mean duration of an excursion below the threshold, *s*_{W}, can be obtained using a crossing analysis as follows (Porporato *et al*. 2001):

- (9)

where *λ*′ is the frequency of rainfall events given by *λ*exp(−Δ/*α*), and *γ*[·,·] is the lower incomplete gamma function.

Porporato *et al*. (2001) defined the measure of vegetation water stress combining with the mean duration and frequency of water stress during the growing season, known as ‘dynamic’ water stress or mean total dynamic stress during the growing season, . Likewise, to describe plant mortality and survival induced by hydraulic failure under severe drought conditions, here we defined a ‘tree death index’, *η*, using Eqns 8 and 9 as follows:

- (10)

where *k* is a parameter representing an index of plant resistance to drought and *T*_{seas} is the length of a growing season. Tree mortality described by *η* was related to a permanent damage caused by drought-induced hydraulic failure. Note that, unlike Porporato *et al*.'s , we use instead of using the mean duration of an excursion below *s**. Moreover, we do not take into account the effect of the frequency of water stress in Eqn 10 because permanent damage leading to tree mortality occurs at the maximum level of water stress and tree mortality is a single event. In this study, the tree death index during a season, *η*, is used to analyse plant mortality and survival of isohydric and anisohydric species in each given climatic condition.